Recall that Huisken has classified the cylinders \(\mathbb {S}^k(\sqrt{2k})\times \mathbb {R}^{n-k}\)  as the unique smooth embedded self-shrinkers with non-negative mean curvature \(H \ge 0\)  [1]}, [2]} (the technical assumption of bounded curvature was later removed by Colding–Minicozzi [3]}). The following result can be viewed as a geometric consequence of Huisken's result. It will serve as our key mechanism for perturbing away “non-generic” singularities.
